Self-testing of symmetric three-qubit states

Fei Gao; Qiaoyan Wen; Xinhui Li; Yukun Wang; Yunguang Han

arxiv: 1907.06397 · v1 · pith:4AH3GUFBnew · submitted 2019-07-15 · 🪐 quant-ph

Self-testing of symmetric three-qubit states

Xinhui Li , Yukun Wang , Yunguang Han , Fei Gao , Qiaoyan Wen This is my paper

Pith reviewed 2026-05-24 21:43 UTC · model grok-4.3

classification 🪐 quant-ph

keywords self-testingsymmetric three-qubit statesW stateGHZ statedevice-independent certificationswap methodsemidefinite programming

0 comments

The pith

Symmetric three-qubit states that superpose the W and GHZ states can be self-tested device-independently from their measurement statistics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes self-testing protocols for the family of symmetric three-qubit states formed as superpositions of the W state and the GHZ state. For the special case of equal coefficients in the computational basis, an analytic criterion proves uniqueness by combining subsystem self-testing on a partially entangled state and a maximally entangled state at the same time. For arbitrary coefficients the same uniqueness is shown numerically to high precision by applying the swap method inside a semidefinite program. The only data used are the number of measurements, the number of outcomes per measurement, and the observed probabilities.

Core claim

We propose self-testing schemes for a large family of symmetric three-qubit states, namely the superposition of W state and GHZ state. We first propose and analytically prove a self-testing criterion for the special symmetric state with equal coefficients of the canonical basis, by designing subsystem self-testing of partially and maximally entangled state simultaneously. Then we demonstrate for the general case, the states can be self-tested numerically by the swap method combining semi-definite programming (SDP) in high precision.

What carries the argument

Analytic self-testing criterion obtained by simultaneous subsystem self-testing of a partially entangled two-qubit state and a maximally entangled two-qubit state (equal-coefficient case); swap method inside semidefinite programming (general case).

If this is right

The equal-coefficient symmetric state admits a fully analytic self-testing proof.
Arbitrary superpositions of W and GHZ admit high-precision numerical self-testing.
Self-testing now covers a continuous family of symmetric three-qubit states rather than isolated examples such as pure Dicke or graph states.
Certification requires only the number of measurements, the number of outputs, and the observed statistics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Device-independent protocols that consume these states as resources could be certified without trusting the state-preparation devices.
The same combination of analytic subsystem arguments and numerical SDP may extend self-testing to other continuous families of symmetric multipartite states.
Exact analytic criteria found for special coefficient values can serve as benchmarks for checking the accuracy of numerical SDP relaxations on nearby states.

Load-bearing premise

The observed statistics are assumed to arise exactly from the claimed projective measurements performed on the target symmetric state.

What would settle it

Exhibiting a different three-qubit state (or a higher-dimensional state) together with measurements that produce exactly the same joint probability tables would falsify the uniqueness claim.

Figures

Figures reproduced from arXiv: 1907.06397 by Fei Gao, Qiaoyan Wen, Xinhui Li, Yukun Wang, Yunguang Han.

**Figure 3.** Figure 3: FIG. 3. (Color Online) The plot of lower bounds on the fi [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

Self-testing refers to a device-independent way to uniquely identify the state and the measurement for uncharacterized quantum devices. The only information required comprises the number of measurements, the number of outputs of each measurement, and the statistics of each measurement. Earlier results on self-testing of multipartite state were restricted either to Dicke states or graph states. In this paper, we propose self-testing schemes for a large family of symmetric three-qubit states, namely the superposition of W state and GHZ state. We first propose and analytically prove a self-testing criterion for the special symmetric state with equal coefficients of the canonical basis, by designing subsystem self-testing of partially and maximally entangled state simultaneously. Then we demonstrate for the general case, the states can be self-tested numerically by the swap method combining semi-definite programming (SDP) in high precision.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Analytic self-test for the equal-coefficient W-GHZ state, but only numerical SDP evidence for the general family.

read the letter

The main thing here is an analytic self-testing proof for the equal-coefficient case of the W-GHZ superposition on three qubits, plus numerical SDP checks for the rest of the family. That extends the known self-testing results beyond Dicke and graph states, which is the actual addition. The analytic part works by running simultaneous self-tests on partially entangled and maximally entangled subsystems, and the description suggests the derivation is direct and uses standard swap operators. That piece looks usable on its own. The numerical part for generic coefficients relies on the swap method plus SDP solved to high precision. This gives practical evidence that the target states are recovered, but SDP relaxations are outer approximations and floating-point output never rules out other states or measurements that match within tolerance. So the uniqueness claim for the full continuous family is not rigorously established. The paper stays within standard quantum mechanics and does not introduce fitted parameters or circular reductions. No load-bearing gaps in the cited prior work on Dicke or graph states are visible. Readers working on device-independent certification of multipartite states will find the special-case proof and the SDP setup worth looking at. The work is coherent on its own terms and shows clear engagement with the self-testing literature. It is worth sending to peer review so referees can check the analytic steps in detail and see whether the numerical results can be strengthened or reframed as evidence rather than proof.

Referee Report

1 major / 0 minor

Summary. The paper proposes self-testing schemes for a large family of symmetric three-qubit states, namely superpositions of the W state and GHZ state. It analytically proves a self-testing criterion for the special case with equal coefficients by simultaneous subsystem self-testing of partially and maximally entangled states. For general coefficients, it demonstrates self-testing numerically via the swap method combined with semi-definite programming (SDP) in high precision.

Significance. The analytic proof for the equal-coefficient special case is a strength, as it supplies a rigorous uniqueness result grounded in standard quantum mechanics and self-testing definitions without free parameters or fitted quantities. This extends self-testing beyond the previously considered Dicke and graph states. The numerical SDP results for the general family provide supporting evidence of practical utility but do not rise to the level of a uniqueness proof.

major comments (1)

[Abstract] Abstract: The numerical demonstration for general coefficients via the swap method and SDP in high precision does not establish rigorous uniqueness. SDP relaxations are outer approximations to the set of quantum correlations, and floating-point results are never exactly tight; therefore high-precision numerics alone cannot exclude the existence of other states or measurements that reproduce the observed statistics within the reported tolerance. This concern is load-bearing for the central claim that covers the large family of states.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We appreciate the positive assessment of the analytic proof for the equal-coefficient case. We agree that the numerical SDP results constitute supporting evidence rather than a rigorous uniqueness proof, and will revise the manuscript to clarify this distinction in the abstract and discussion.

read point-by-point responses

Referee: [Abstract] Abstract: The numerical demonstration for general coefficients via the swap method and SDP in high precision does not establish rigorous uniqueness. SDP relaxations are outer approximations to the set of quantum correlations, and floating-point results are never exactly tight; therefore high-precision numerics alone cannot exclude the existence of other states or measurements that reproduce the observed statistics within the reported tolerance. This concern is load-bearing for the central claim that covers the large family of states.

Authors: We agree with the referee. The SDP-based numerical demonstration supplies high-precision evidence but cannot rigorously establish uniqueness, as correctly noted regarding relaxations and floating-point precision. In the revised manuscript we will update the abstract to state that the general family is supported by numerical evidence via the swap method and SDP (rather than claiming it establishes self-testing with the same rigor as the analytic case), and we will add a brief discussion of this limitation in the conclusions. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical construction and numerical SDP are independent of target state by definition.

full rationale

The manuscript derives its self-testing criterion for the equal-coefficient case via an explicit analytical construction of simultaneous subsystem self-testing on partially and maximally entangled states; this is a direct proof from observed correlations and does not reduce to the target state by construction or by fitting. The general-coefficient case is handled by the standard swap-method SDP relaxation, which is an external numerical verification tool rather than a self-referential fit or self-citation. No load-bearing self-citation, ansatz smuggling, or renaming of known results appears; the derivation chain remains self-contained against standard quantum-information benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard axioms of quantum mechanics and the operational definition of self-testing; no additional free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Quantum mechanics with projective measurements and the operational definition of self-testing from statistics alone.
Invoked throughout the abstract as the foundation for the proposed criteria.

pith-pipeline@v0.9.0 · 5672 in / 1029 out tokens · 17095 ms · 2026-05-24T21:43:35.850649+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 2 internal anchors

[1]

0" or "1

with diﬀerent kinds of coeﬃcients. A. Self-testing of a symmetric three-qubit state The speciﬁc case we consider is the state with equal coeﬃcient of the basis, reads as: |ψ ⟩ = 1√ 5 (|001⟩ + |010⟩ + |100⟩ + |000⟩ + |111⟩). (9) The basic idea of self testing this state is to project the state onto two kinds of subsystem entangled states by one FIG. 1. The...

work page
[2]

(35) Observation (16a) implies ⟨P 1 CA2(B2 − B1)⟩ = 0 ⇒ P 1 CX ′ A |ψ ⟩ ⊥ P 0 CZ ′ B |ψ ⟩ (36) and combine the ﬁrst relation in ( 34), we have P 1 CX ′ A |ψ ⟩ ⊥ P 1 CZ ′ A |ψ ⟩

implies P 1 CZ ′ A |ψ ⟩ =P 1 CZ ′ B |ψ ⟩, P 1 CX ′ A |ψ ⟩ =P 1 CX ′ B |ψ ⟩ (34) and the anti-commutation relations P 1 CZ ′ AX ′ A |ψ ⟩ = −P 1 CX ′ AZ ′ A |ψ ⟩, P 1 CZ ′ BX ′ B |ψ ⟩ = −P 1 CX ′ BZ ′ A |ψ ⟩. (35) Observation (16a) implies ⟨P 1 CA2(B2 − B1)⟩ = 0 ⇒ P 1 CX ′ A |ψ ⟩ ⊥ P 0 CZ ′ B |ψ ⟩ (36) and combine the ﬁrst relation in ( 34), we have P 1 CX ...

work page
[3]

This state can be normalized into the form of |junk⟩ABC ⊗ |ψ ⟩A′B′C ′, here |junk⟩ = √ 5P 0 AP 0 BP 0 C |ψ ⟩

to |Ψ′⟩ =P 0 AP 0 BP 0 C |ψ ⟩ (|000⟩ + |001⟩ + |010⟩ + |100⟩ + |111⟩). This state can be normalized into the form of |junk⟩ABC ⊗ |ψ ⟩A′B′C ′, here |junk⟩ = √ 5P 0 AP 0 BP 0 C |ψ ⟩. Thus, we have proven that, with these requirements (15)–( 21) on the measurement results indeed self-test the unknown state as target state ( 9). Further more, we also consider...

work page
[4]

localizing ma- trix

and ( 48), where Γ is a matrix with NPA hierarchy characterization of the quantum behaviors. This moment matrix corre- sponding to q-local level 1 (which includes any products with at most one operator per party) has size 74 × 74 7 and is augmented by necessary terms(like ⟨ZAXAZA⟩, ⟨ZBXBZCXC⟩, ⟨ZCXCZAXAZA⟩ and so on) to express all the average values ⟨·⟩t...

work page
[5]

(57) Deﬁne |j′ A⟩ ≡ ∑ j Uji |j⟩, |k′ B⟩ ≡ ∑ k Vik |k⟩ and λ i = Sii, for i,j,k ∈ { 0, 1}

0 0 √ 1 6 (3 − √ 5)   (56) is diagonal, U and V are unitary matrices: U =   3+ √ 5 2 √ 5+2 √ 5 3− √ 5 2 √ 5− 2 √ 5 1+ √ 5 2 √ 5+2 √ 5 1− √ 5 2 √ 5− 2 √ 5  , V =   1+ √ 5 2 √ 5+ √ 5 1− √ 5√ 10− 2 √ 5√ 2 5+ √ 5 5+ √ 5 10  . (57) Deﬁne |j′ A⟩ ≡ ∑ j Uji |j⟩, |k′ B⟩ ≡ ∑ k Vik |k⟩ and λ i = Sii, for i,j,k ∈ { 0, 1}. Obviously, {|j′⟩}A and {|k′⟩}B are ...

work page
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Bennett C H, Brassard G, Crépeau C, et al. Teleport- ing an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical review let- ters, 1993, 70(13): 1895

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[1] [1]

0" or "1

with diﬀerent kinds of coeﬃcients. A. Self-testing of a symmetric three-qubit state The speciﬁc case we consider is the state with equal coeﬃcient of the basis, reads as: |ψ ⟩ = 1√ 5 (|001⟩ + |010⟩ + |100⟩ + |000⟩ + |111⟩). (9) The basic idea of self testing this state is to project the state onto two kinds of subsystem entangled states by one FIG. 1. The...

work page

[2] [2]

(35) Observation (16a) implies ⟨P 1 CA2(B2 − B1)⟩ = 0 ⇒ P 1 CX ′ A |ψ ⟩ ⊥ P 0 CZ ′ B |ψ ⟩ (36) and combine the ﬁrst relation in ( 34), we have P 1 CX ′ A |ψ ⟩ ⊥ P 1 CZ ′ A |ψ ⟩

implies P 1 CZ ′ A |ψ ⟩ =P 1 CZ ′ B |ψ ⟩, P 1 CX ′ A |ψ ⟩ =P 1 CX ′ B |ψ ⟩ (34) and the anti-commutation relations P 1 CZ ′ AX ′ A |ψ ⟩ = −P 1 CX ′ AZ ′ A |ψ ⟩, P 1 CZ ′ BX ′ B |ψ ⟩ = −P 1 CX ′ BZ ′ A |ψ ⟩. (35) Observation (16a) implies ⟨P 1 CA2(B2 − B1)⟩ = 0 ⇒ P 1 CX ′ A |ψ ⟩ ⊥ P 0 CZ ′ B |ψ ⟩ (36) and combine the ﬁrst relation in ( 34), we have P 1 CX ...

work page

[3] [3]

This state can be normalized into the form of |junk⟩ABC ⊗ |ψ ⟩A′B′C ′, here |junk⟩ = √ 5P 0 AP 0 BP 0 C |ψ ⟩

to |Ψ′⟩ =P 0 AP 0 BP 0 C |ψ ⟩ (|000⟩ + |001⟩ + |010⟩ + |100⟩ + |111⟩). This state can be normalized into the form of |junk⟩ABC ⊗ |ψ ⟩A′B′C ′, here |junk⟩ = √ 5P 0 AP 0 BP 0 C |ψ ⟩. Thus, we have proven that, with these requirements (15)–( 21) on the measurement results indeed self-test the unknown state as target state ( 9). Further more, we also consider...

work page

[4] [4]

localizing ma- trix

and ( 48), where Γ is a matrix with NPA hierarchy characterization of the quantum behaviors. This moment matrix corre- sponding to q-local level 1 (which includes any products with at most one operator per party) has size 74 × 74 7 and is augmented by necessary terms(like ⟨ZAXAZA⟩, ⟨ZBXBZCXC⟩, ⟨ZCXCZAXAZA⟩ and so on) to express all the average values ⟨·⟩t...

work page

[5] [5]

(57) Deﬁne |j′ A⟩ ≡ ∑ j Uji |j⟩, |k′ B⟩ ≡ ∑ k Vik |k⟩ and λ i = Sii, for i,j,k ∈ { 0, 1}

0 0 √ 1 6 (3 − √ 5)   (56) is diagonal, U and V are unitary matrices: U =   3+ √ 5 2 √ 5+2 √ 5 3− √ 5 2 √ 5− 2 √ 5 1+ √ 5 2 √ 5+2 √ 5 1− √ 5 2 √ 5− 2 √ 5  , V =   1+ √ 5 2 √ 5+ √ 5 1− √ 5√ 10− 2 √ 5√ 2 5+ √ 5 5+ √ 5 10  . (57) Deﬁne |j′ A⟩ ≡ ∑ j Uji |j⟩, |k′ B⟩ ≡ ∑ k Vik |k⟩ and λ i = Sii, for i,j,k ∈ { 0, 1}. Obviously, {|j′⟩}A and {|k′⟩}B are ...

work page

[6] [6]

Quantum entanglement

Horodecki R, Horodecki P, Horodecki M, et al. Quantum entanglement. Reviews of modern physics, 2009, 81(2): 865

work page 2009

[7] [7]

Teleport- ing an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels

Bennett C H, Brassard G, Crépeau C, et al. Teleport- ing an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical review let- ters, 1993, 70(13): 1895

work page 1993

[8] [8]

Classical command of quantum systems

Reichardt B W, Unger F, Vazirani U. Classical command of quantum systems. Nature, 2013, 496(7446): 456

work page 2013

[9] [9]

Public quantum com- munication and superactivation

Brandao F G S L, Oppenheim J. Public quantum com- munication and superactivation. IEEE Transactions on Information Theory, 2012, 59(4): 2517-2526

work page 2012

[10] [10]

Scalable quantum search using trapped ions

Ivanov S S, Ivanov P A, Linington I E, et al. Scalable quantum search using trapped ions. Physical Review A, 9 2010, 81(4): 042328

work page 2010

[11] [11]

Spin state tomog- raphy of optically injected electrons in a semiconductor

Kosaka H, Inagaki T, Rikitake Y, et al. Spin state tomog- raphy of optically injected electrons in a semiconductor. Nature, 2009, 457(7230): 702

work page 2009

[12] [12]

The device-independent outlook on quantum physics

Scarani V. The device-independent outlook on quantum physics. Acta Physica Slovaca, 2012, 62(4): 347-409

work page 2012

[13] [13]

Proposed ex- periment to test local hidden-variable theories

Clauser J F, Horne M A, Shimony A, et al. Proposed ex- periment to test local hidden-variable theories. Physical review letters, 1969, 23(15): 880

work page 1969

[14] [14]

Some results and problems on quan- tum Bell-type inequalities

Tsirelson B S. Some results and problems on quan- tum Bell-type inequalities. Hadronic Journal Supple- ment, 1993, 8(4): 329-345

work page 1993

[15] [15]

Self testing quantum apparatus

Mayers D, Yao A. Self testing quantum apparatus. arXiv preprint quant-ph/0307205, 2003

work page internal anchor Pith review Pith/arXiv arXiv 2003

[16] [16]

Robust self-testing of unknown quantum systems into any entangled two-qubit states

Yang T H, Navascués M. Robust self-testing of unknown quantum systems into any entangled two-qubit states. Physical Review A, 2013, 87(5): 050102

work page 2013

[17] [17]

Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalitie s and their application to self-testing

Bamps C, Pironio S. Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalitie s and their application to self-testing. Physical Review A, 2015, 91(5): 052111

work page 2015

[18] [18]

All pure bipartite entangled states can be self-tested

Coladangelo A, Goh K T, Scarani V. All pure bipartite entangled states can be self-tested. Nature communica- tions, 2017, 8: 15485

work page 2017

[19] [19]

Optimal robust quantum self-testing by binary nonlocal XOR games

Miller C A, Shi Y. Optimal robust quantum self- testing by binary nonlocal XOR games. arXiv preprint arXiv:1207.1819, 2012

work page internal anchor Pith review Pith/arXiv arXiv 2012

[20] [20]

All the self-testings of the singlet for two binary measurements

Wang Y, Wu X, Scarani V. All the self-testings of the singlet for two binary measurements. New Journal of Physics, 2016, 18(2): 025021

work page 2016

[21] [21]

Robust self-testing of the singlet

McKague M, Yang T H, Scarani V. Robust self-testing of the singlet. Journal of Physics A: Mathematical and Theoretical, 2012, 45(45): 455304

work page 2012

[22] [22]

Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mer- min inequalities

Kaniewski J. Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mer- min inequalities. Physical review letters, 2016, 117(7): 070402

work page 2016

[23] [23]

Physical char- acterization of quantum devices from nonlocal correla- tions

Bancal J D, Navascué M, Scarani V, et al. Physical char- acterization of quantum devices from nonlocal correla- tions. Physical Review A, 2015, 91(2): 022115

work page 2015

[24] [24]

Robust and versa- tile black-box certiﬁcation of quantum devices

Yang T H, Vértesi T, Bancal J D, et al. Robust and versa- tile black-box certiﬁcation of quantum devices. Physical review letters, 2014, 113(4): 040401

work page 2014

[25] [25]

Self-testing graph states

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